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## equation with critical growth

1*

1

### and Huiwen Yan

2

*Correspondence:

yangzhou1975@yahoo.com.cn 1School of Mathematics Science, South China Normal University, Guangzhou, 510631, China Full list of author information is available at the end of the article

Abstract

The existence of a positive solution of ap-Laplace-like equation with critical growth is established by the generalization to the concentration-compactness principle and the Sobolev inequality under some proper assumptions. Moreover, we achieve some regularity results of the solution.

MSC: 35J65

Keywords: p-Laplace-like operator; critical growth; concentration-compactness

principle; weakly continuity

1 Introduction

This paper is devoted to the existence of a positive solution of the followingp-Laplace-like problem with critical growth:

⎧ ⎨ ⎩

–div(a(∇u)) =f(x,u), x,

u= , x, (.)

whereis a smooth bounded domain inRN,  <p<N, and the functionsa,fsatisfy some

proper conditions, the details of which are described later.

There were many papers about the existence of the solution ofp-Laplacian problems involving critical growth such as [–]. In them,a(ξ) =|ξ|p–ξ andf are some concrete functions with critical growth, which means thatf(x,u)|u|–p* does not converge to zero

asu→ ∞, wherep*is the critical exponent, i.e.,p*=Np/(Np). The concentration-compactness principle, which was built by Lions in [, ], plays an important role in achiev-ing the existence of a nontrivial solution of the problems in them.

The authors proved the existence of a nonnegative and nontrivial solution for a Dirich-let problem forp-mean curvature operate with critical growth in [], wherea(ξ) = ( + |ξ|)p– ξ,p, andf is some concrete function involving a critical exponent. Since the

functionahas an explicit form, the authors can use the concentration-compactness prin-ciple to achieve their results, too. But ifais an abstract function in problem (.), then the problem becomes more complicated and interesting, and Lions’ C-C principle cannot be directly applied to it. Thanks to the generalization of the C-C principle in [], we can establish the existence of a nonnegative and nontrivial solution of equation (.) if we im-pose some proper conditions on the functionsaandf and make more careful estimates. Moreover, we achieve some regularity result of the solution and prove the solution is posi-tive under some proper assumptions. The results can be easily extended to a more general

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p-Laplace-like equation with critical growth and singular weights by the Caﬀarelli-Kohn-Nirenberg inequality and the method in [].

Recently, there have been some articles on stochastic partial diﬀerential equations (SPDEs) involvingp-Laplace operator; see [, ]. Some estimates and properties of the solution of the corresponding elliptic equations are important to the research on SPDEs. So, the results in this paper may be useful in the study ofp-Laplace SPDEs with critical growth.

In this paper, we suppose that the potentiala:RNRNsatisﬁes the following

assump-tions.

LetA=A(ξ) :RNRbe of continuous derivative with respect toξ witha=∇ξAand

satisfy the following conditions: (A) A() = ,

(A) there arep> andm∈[,p], three positive constantsa,aandasuch that

a|ξ|pA(ξ)≤a|ξ|p+a|ξ|m for allξRN,

(A) A(ξ)is strictly convex inξ, that is,A(ξ+η) <A(ξ) +A(η)for anyξ=ηRN,

(A) there exists a positive numberasuch thatlim|ξ|→+∞a(ξξ|ξ|–p=a.

We impose some assumptions on the critical nonlinear termf(x,u) :×RR, which is continuous, as follows:

(B) f(x, ) = ,

(B) there is a functionb(x)L∞()such that

lim sup

u→ sup

x∈

f(x,u)u|u|–pb(x),

(B) there are two positive numberscandcsuch that

lim inf

u→∞ x∈inf

f(x,u)u|u|–p*≥c, lim sup

u→∞ supx∈

f(x,u)u|u|–p*≤c,

(B) denoteF(x,u) =uf(x,s)ds, which satisﬁes

lim inf

u→∞ x∈inf

f(x,u)up*F(x,u)|u|–p*≥.

Moreover, we supposeaandf satisfy the next correlation. (C) there exists aβ> such that

pA(υ) –b(x)|υ|pdxβ

A(υ)dx for anyυX,

whereXisW,p(),i.e., the completion ofC∞()with the norm u X= (

|∇u|

pdx)/p.

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in another paper. Sincea(z) is not an explicit function, we need to use the generalization of the C-C principle in [] and more subtle estimates to study problem (.).

It is clear that the solution of problem (.) is the critical point of the variational func-tional

I(u) =

A(u)dx

F(x,u)dx. (.)

Moreover,I(u) is continuous diﬀerentiable inX, and its Fréchet derivation is

I(u),υ=

a(∇u)· ∇υdx

f(x,u)υdx,υX. (.)

The ﬁrst main result in this paper is

Theorem . Suppose problem(.)satisﬁes assumptions(A)-(A), (B)-(B)and(C).

Moreover,there exists a nontrivialυ∈X such thatυ≥and

(C) supt≥I(tυ) < (pp*)(pS)

p* p*–pc

p pp*

 ,whereS=infυ∈X\{} A(∇υ)  υpp*,with

υ qq=

|υ|qdx(q≥).

Then problem(.)has a nonnegative and nontrivial solution.

Since the condition (C) is diﬃcult to check, we give another easily checked theorem.

Theorem . Assume conditions(A)-(A), (B)-(B)and(C)are satisﬁed and

(A) pA(ξ)≥a|ξ|pfor anyξRN,

(B) there exists a nonempty setWsuch thatF(x,u) –c

p*|u|p *

≥for any

xW,uR, (C) limε→+(ε

(Np)m

(p–)pN+K

(φ))(K(ψ))–= ,whereK(ψ) =

ε

(r)dr, K(φ) =

ε

(r)drwith

(r) =ψ

ε–  +rp/(p–)

Np

p

rN–, ψ(r) = min

u≥r,x∈W

F(x,u) –cp*|u|

p*

,

(r) =φ

ε–

 +rp/(p–)

N p

r/(p–)

rN–, φ(r) = max ≤|ξ|≤r

pA(ξ) –a|ξ|p

.

Then problem(.)has a nonnegative and nontrivial solution in X.

To establish the regularity of the solutionuand proveu>  in, we need to impose stronger assumptions on the potentialaand the nonlinear termf, which are as follows:

(D) A(ξ)∈C(RN\ {})witha() =and there exist positive numberscandC,

κ∈[, ]such that for anyξRN\ {}andηRN,

N

i,j=

A

∂ξi∂ξjηiηjc

κ+|ξ|p–|η|,

N

i,j=

A

∂ξi∂ξj

+|ξ|p–,

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Theorem . Assume the assumptions in Theorem.or those in Theorem.hold,then there exists a constantα∈(, )such that the solution uC,α()if assumption(D)holds.

Moreover,if assumption(D)is satisﬁed,then u> in.

In Section , we will prove the main results. Some corollaries and examples are shown in Section .

2 The proof of the main results

First, we present the main tool in this paper - the generalized concentration-compactness principle, which is easily deduced from Theorem . in [].

Lemma . Suppose assumptions(A), (A)and(A)hold,unweakly converges to u in

X and μn=A(∇un)dx,νn=|un|p

*

dx converge toμ, νweakly in the sense of measures, respectively.

Then there exist some at most countable set J,a family{xj;jJ}of distinct points in,

and two families{νj;jJ},{μj;jJ}of positive numbers such that

ν=|u|p*dx+

j∈J

νjδxj, μA(∇u)dx+

j∈J μjδxj,

μjS(νj)p/p

*

(∀jJ),

(.)

whereδxjdenotes the Dirac measure at the point xj.

Second, we deduce some properties ofaandf by the similar method as in [] or []. According to assumptions (A)-(A) and (B)-(B), we conclude that for anyσ> ,pkp*, there exist some constantsC

k,σ,,Csuch that for anyξRn,x,uR,

pA(ξ) –a|ξ|p+a(ξξa|ξ|pσ|ξ|p+, (.)

c p*|u|

p*CF(x,u)b(x) +σ

p |u|

p+c+σ p* |u|

p*+C

k,σ|u|k, (.)

c |u|

p*C|u| ≤f(x,u)ub(x) +σ|u|p+ (c

+σ)|u|p

*

+Ck,σ|u|k, (.)

f(x,u)up*F(x,u)≥–σ|u|p*–,

 +σ c

f(x,u)up*F(x,u)≥–Cσ.

(.)

To obtain a nonnegative solution, we ﬁrst consider the following variational functional and its Fréchet derivation:

I(u) =

A(u)dx

Fx,u+dx; (.)

I(u),υ=

a(∇u)· ∇υdx

fx,u+υdx,υX. (.)

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Lemma .I() = and there exist two constantsα,ρand a function u∈X such that

I(u)|Bρ

()≥α> , u X>ρ, I(u)≤, (.)

where Bρ() ={uX, u Xρ},∂Bρ()denotes the boundary of Bρ().

Proof Letu=  in (.), and we haveI() = . Choosingk=p*,σ=a

βS/ in (.) (whereS=infυ∈X\{} υ pX υ

p

p* is the best

embed-ding constant fromXtoLp*()), and combining assumptions (C), (A), we have

I(u)

A(u)dx

b(x) +aβS

p u

+p

+Cu+p*dxaβ p u

p XC u

p* X.

Hence,I(u) >  if u Xis small enough. So, we have showed the existence ofαandρin (.).

Next, we constructusatisfying (.). In fact, ﬁxing a nonnegative and nontrivial func-tionu, recalling assumption (A) and (.), we deduce that there are positive constants C,C,Csuch that

I(tu)≤

atp|∇u|p+atm|∇u|m

dx

ctp

*

p* u +p*+C

dx

CtpCtp

*

+C.

Hence, iftis large enough, then we can setu=tusatisfying (.).

According to the Ambrosetti-Rabinowitz mountain pass theorem without the (PS) con-dition, there exists a function sequence{un}∞n=⊂Xsuch that asn→ ∞,

I(un) –→c and I(un) –→ inX*, wherec=inf

γmaxu∈γI(u)α> , (.)

denotes the class of continuous paths joining  tou inX,X*denotes the dual space ofX.

Lemma . The sequence{un}∞n=is bounded in X.

Proof Letυ=unin (.) and combine (.), (.). We see that asn→ ∞,

c+o() =I(un) –δI(un),un

= ( –)

A(un)dx

+δ

pA(∇un) –a(∇un)· ∇un

+

fx,u+nun

δF

x,u+ndx

a( –) – δσ

un pX, whereδ=

p*

 +σ c

.

In the last inequality, we have used (.) and (.). If we ﬁx a small enoughσ such that a( –) – δσ>  in the above inequality, then the conclusion in this lemma is obvious.

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As a result of the above preparations, we can prove Theorem ..

Proof of Theorem. Since{un}∞n=is bounded inX, it is easy to see there are auXand a subsequence of{un}∞n=, still denoted by itself , such that

unu weakly inX,

unu a.e. inand strongly inLq()

≤q<p*,

(.)

fx,u+nfx,u+ weakly inX*. (.)

By the Helly theorem, there exist a subsequence, still denoted by itself, and two nonneg-ative measuresμandνsuch that asn→ ∞,

A(∇un)dx w

–→μ, |un|p

*

dx–→w ν weakly in the sense of measures. (.)

Applying Lemma ., we have the corresponding conclusions of Lemma .. Next, we establish the lower-bound ofνjandμj

νj≥(pS)p

*/(p*p)

cp*/(pp)*> , μjpp/(p

*p)

Sp*/(p*–p)cp/(pp*). (.)

Denoteϕas the cutoﬀ function of the ballB() inRN,i.e., which satisﬁes

ϕCRN, ≤ϕ≤,

ϕ(x) =  ifxB(), ϕ(x) =  ifxB().

(.)

Deﬁneϕε,j=ϕ((x–xj)/ε) for everyε>  andxjRN for anyjJ. Recalling Lemma  in

[], we have the following estimate:

|unϕε,j|pdxS– RN|∇

ϕ| p*p p*–pdx

p*–p p*

B(xj,ε)

|∇un|pdx

C

B(xj,ε)

|∇un|pdx. (.)

Hence,{unϕε,j}∞n=is still bounded inXand the boundary is independent ofε,j.

Letu=un,υ=unϕε,jin (.) and combine (.), (.), (.), (.). Then we obtain as

n→ ∞,

o() =I(un),unϕε,j

=

a(∇un)·(∇unϕε,j+unϕε,j) –f

x,u+nϕε,jundx

p

/(xj)

A(un)dx– σ un pXmes

()

(xj)

(c+σ)|un|p

*

+

dx

a(∇un) p p–dx

p–

p

C

B(xj,ε)

|∇un|pdx+o()

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In the above inequality, ﬁrst lettingn→ ∞, then takingε→+, and ﬁnally takingσ+, we deducepμjcνj. So, (.) implies (.). Sinceνis a bounded measure,Jis at most

ﬁnite.

In the following, we prove thatu≥, and it is the solution of problem (.). IfJis empty, then the proof is similar to that whenJis nonempty, which we omit. Next, we suppose J is nonempty and denoteJ={, , . . . ,m},ε={x∈|dist(x,xj) >ε,∀j∈J}. Fix a large

enoughRand a suﬃciently smallεso that

¯

BR(), (xi)∩(xj) =Ø whilei=j and

m

j=

(xj)⊂BR().

Deﬁneψε(x) =ϕ(x/R) –

m

j=ϕε,j(x) withxRN,  <εε. It is not diﬃcult to deduce that{ψεun}∞n=is bounded inXand the bound is independent ofεfrom (.). According to (.) and (.), it is clear that asn→ ∞,

o() =I(un) –I(u), (unu)ψε

=

J(un,u)ψε+J(un,u) +J(un,u)dx, (.)

whereJ(un,u) = (a(un) –a(u))·(∇un–∇u),J(un,u) = (a(un) –a(u))· ∇ψε(unu)

andJ(un,u) = (f(x,u+) –f(x,u+n))ψε(u–un). Applying the method as in (.), we see that

|J(un,u)dx|converges to  asn→ ∞. Moreover, the deﬁnition ofψεand (.) imply

lim

n→∞ |unψε| p*dx=

|ψε|p

*

=

|uψε|p

*

dx.

Recalling (.), we seeunψεuψεinLp

*

(). In view of (.), we obtain

J(un,u)dx

f

x,u++fx,u+n

p* p*–dx

p*–

p* ψ

ε(unu)p*→. (.)

According to (.), (.), we deduce thatJ(un,u) converges to  a.e. inasn→ ∞,

maybe extracting a subsequence. Since A(ξ) is strictly convex, by the same method as [], we claim∇un→ ∇ua.e. in, and there exists a subsequence, still denoted by itself,

such thata(∇un) weakly converges toa(∇u) inX*. Hence, (.), (.) and (.) imply that I(un) weakly converges toI(u) inX*andI(u) = . So, it is not diﬃcult to see thatu≥

andI(u) =I(u) = , which means thatuis a weakly solution of equation (.).

Next, we proveuis nontrivial,i.e., u=  if assumption (C) holds. According to the deﬁnition ofI(u) and the properties of{un}∞n=, we conclude that asn→ ∞,

o() +c–I(u)

–  p*

j∈J

I(un),unϕε,j

=

A(∇un) –A(∇u)

dx

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–  p*

j∈J

a(∇un)· ∇(unϕε,j) –f

x,u+nϕε,jundx

=K+K+  p*

j∈J

p*K,j+

p*–p

A(∇un)ϕε,jdx+K,jK,j+K,j

, (.)

whereK,j=

(F(x,u

+) –A(∇u))ϕ

ε,jdxconverges to  asε→+, and it has been proved

in (.) thatK,j=

a(∇un)· ∇ϕε,jundxconverges to  asn→ ∞. Moreover, asn→ ∞,

K=

A(∇un) –A(∇u)

ψεdx

=

ψεdμ

ψεA(u)dx+o()o()

by (.),

K=

Fx,u+nFx,u+ψεdx=o()

by the method similar to that in (.),

K,j=

pA(un) –a(un)· ∇un

ϕε,jdx

≥–σ un pXmes

() by (.)

,

K,j=

fx,u+nunp*F

x,u+nϕε,jdx

≥–σ un p

*

p*–mes

() by ( .)

.

Combining (.), (.) and the above inequalities and equalities, ﬁrstly takingn→ ∞, then takingε→+, and ﬁnally takingσ+in (.), we have

c–I(u) =c–I(u)p*–p

p* μj

p

p*

(pS)

p* p*–pc

p pp*

 .

According to the deﬁnition of cand assumption (C), we haveI(u) < , which means

u= .

Before proving Theorem ., we need to introduce the function family{υε}which

ap-proximates the best embedding constantSfromXtoLp*(). Without loss of

generaliza-tion, we supposeB()⊂,  <ε< . Denote

U= 

( +|x|p/(p–))(Np)/p, =ε pN

p U

x

ε

, =Uεϕ, υε= –p*,

whereϕis deﬁned in (.), andUis the extremal function reachingS. It is easy to check that asε→+,

RN\B

()

|Uε|p

*

dx=OεpN–,

RN\B

()

|∇Uε|pdx=O

ε

Np

p–, (.)

B()\B()

||mdx=

B()\B()

|∇|mdx=O

ε

(Np)m

(p–)p, (.)

υε p

*

p*= , υε pp=o(), υε pX=S+O

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As a result of the preparations, we can prove Theorem . as follows.

Proof of Theorem. Without loss of generalization, we supposeB()⊂W. For conve-nience, let AandBdenote some positive constants which may be diﬀerent in diﬀerent places.

Applying the method in the proof of Lemma ., we deduce thatI(tυε)→–∞ast

+∞, which implies there exists a≥ such thatI(tευε) =supt≥I(tυε) and

d

dtI(tυε)|t==

I(tευε),υε

=

a(tευε)· ∇υεdx

f(x,tευε)υεdx= . (.)

Letσ=  andk=pin (.), combining (.), (A), (A) and (A), we obtain

atεp υε pX

A(tευε)dx

a(tευε)· ∇υεdx

Ctpε υε pp+tp

*

ε υε p

*

p*

.

Recalling (.), we see thatis positive and bounded away from zero asε→+.

More-over, according to assumption (A) and (.), we infer

Ctεp υε pX+ 

a(tευε)· ∇υεdx=

f(x,tευε)tευεdx

c t

p*

ε υε p

*

p*–C.

Hence, (.) impliesis bounded, and the bound is independent ofε.

Set

h(t) = a

pt

p υ

ε pX

cp*t

p* υ

ε p

*

p*.

In view of (.), we have

max

t≥h(t) =

p–  p*

aυε pX

υε pp*

p* p*–p

c

p pp*

 =  p–  p*

(aS)

p* p*–pc

p pp*

 +O

ε

Np p–.

It is noted thatφ(r) is increasing withφ() =  according to assumption (C). Hence, we deduce

sup

t≥

I(tυε) =I(tευε)≤h(tε) +

p

φtε|∇υε|

dx

ψ(tευε)dx

≤  p–  p*

(aS)

p* p*–pc

p pp*

+ Np

p–+

p

φtε|∇υε|

dx

ψ(tευε)dx. (.)

Next, we handle the last two terms on the right-hand side of (.). Sinceφ(r) is nonneg-ative and increasing, we can utilize the properties ofυεandto calculate

≤

φtε|∇υε|

dx

B()

φA|∇Uε|

dx+

B()\B()

φA|∇uε|

(10)

B()

φA|∇Uε|

dx

B/ε()

εNφAεpN|∇U|dx

BεN

ε–

 +

– ε–

(r)dr, (.)

where(r) is deﬁned in assumption (C). Without loss of generalization, supposeA>  and  <ε< . According to the deﬁnition ofφ and assumption (A), it is clear that ≤

φ(r)≤arp+arm, then we have

BεN – ε–

(r)dr=

B()\B()

φA|∇|

dx=

(Np)m

(p–)p. (.)

We have used (.) and (.) in the last equality. Furthermore, (.) and the deﬁnition ofψ(u) imply thatψ(u)≤B(|u|p+|u|p*). Repeating the above argument, we obtain

ψ(tευε)dxBεN ε–

(r)dr+ Np

p–. (.)

Remembering assumption (A), we seepSaS. Combining (.)-(.), we obtain

sup

t≥

I(tυε)≤

p

p*

(pS)

p* p*–pc

p pp*

 –εN

BK(ψ) –BK(φ)

+

(Np)m

(p–)p.

So, assumption (C) implies that assumption (C) holds if we chooseεsmall enough, and

the conclusion in Theorem . follows from Theorem ..

Proof of Theorem. We ﬁrstly proveuL∞() by the Moser iteration. Since the problem involves critical growth, we need some preparation before making the Moser iteration. Set

η(t)∈C(R) and

η(t) =

⎧ ⎨ ⎩

sgn(t)|t|k if|t| ≤M,

linear if|t| ≥M,

wherek> ,M> , ξ(t) =

t

η(s)pds.

It is not diﬃcult to check thatη(t) >  andξ(u)∈XifuX, and for anytR,

ξ(t)≤kη(t)η(t)p–, η(t)≤k +η(t),

ξ(t)|t|p–≤kpη(t)p.

(.)

Letυ(x) =ξ(u(x))ψp(x) in (.), whereψ(x) =ϕ((x–x)/R) andϕis deﬁned in (.), x⊂,R> . DenoteD=BR(x)∩,E=BR(x)∩, then we compute

a(∇u)· ∇(u)ψpdx+p

a(∇u)· ∇ψ ξ(u)ψp–dx

=

(11)

Denote the ﬁrst term and the second term on the left-hand side and on the right-hand side of (.) asJandJ,J, respectively. Now, we estimate them as follows:

J≥a

|∇u|pψpξ(u)dx=a

η(u)pψpdx

=a

D

η(u)ψp–η(u)∇ψpdx,

J≥–pC

|∇u|p–+|∇u|m–ξ(u)|∇ψ|ψp–dx

≥–pkC

D

η(u)∇up–+η(u)p–|∇u|m–η(u)|∇ψ|ψp–dx by (.)

≥–σ D

η(u)pψpdxσJ–Cσkp

D

η(u)p|∇ψ|pdx,

whereσ is a positive number deﬁned later and

J=

D

η(u)p|∇u|mp––pψpdxkp

D

 +η(u)p +|∇u|pψpdx by ( .)

D

η(u)pψpdx+kp

D

η(u)pψpdx+Ckp,

J≤C

|u|p*–+ ξ(u)ψpdx

Ckp

|u|p*–pη(u)p

ψp+η(u) +η(u)p–ψpdx by (.)

Ckp

D

η(u)ψp*dx

p p*

D

|u|p*dx p*–p

p*

+mes(D)

p*–p p*

+Ckp.

Setk=k=p*/pin view of (.) and the above equalities. If we ﬁrstly ﬁx a small enough

σ, then a small enoughR, then we can obtain

E

η(u)p*dx

p/p*

C

E

η(u)pdxC

D

η(u)pdx+C,

whereCis a constant independent ofM. TakingM→+∞, then we deduceuLkp*(E).

Applying a simple covering argument, we achieve thatuLkp*(). Finally, repeating the

same argument, we derive thatuLp*+p(). As a result of the preparations, we can use

the Moser iteration to proveuL∞(). Letυ(x) =ξ(u(x)) in (.), and we calculate

a

η(u)pdx

a(u)· ∇(u)dx=

f(x,u)ξ(u)dx

C

|u|p*–+ ξ(u)dx

Ckp

|u|p*–p+ η(u)p+ dx by (.)

Ckp

η(u)(p*+p)/dx

p p*+p

|u|p*+pdx

p*–p p*+p

+ 

(12)

whereCis a constant independent ofMandk. If we setλ= (p*+p)/(p*), then the above inequality implies

u kp*≤C/kpk/k

 + u kpλkp*

/(kp) .

Thus,uL∞() follows from the standard Moser iteration method.

Applying Theorem  in [], we see that there exists a constant α∈(, ) such that uC,α(). If we rewritef(x,u) as (f(x,u)/u)u, thenf(x,u)/uL() withf(x,u)/u.

Employing Theorem  in [], it is obvious thatu>  in.

3 Some corollaries and examples

In this section, we ﬁrstly consider when (C) is true through analyzingK(ψ) andK(φ), then we give some concrete examples and corollaries.

Firstly, we analyze the eﬀect ofa(z) toK(φ):

Lemma . Suppose a(z)satisﬁes assumptions(A)-(A)and

(A) There exist positive numbersm,m,A,Bsuch thatφ(r)≤Brmfor any≤rA

andφ(r)≤Brmfor anyrA.

Then K(φ) =O(λ(ε) +λ(ε))asε→+,where K(φ)is deﬁned in Theorem.and

λ(ε) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ε

(Np)m

(p–)pN if m

<NN(p––),

ε

mN

p |lnε| if m

=NN(p––),

ε N(–p)

(N–)p if m

>NN(p––),

λ(ε) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ε

N(–p)

(N–)p if m

<NN(p––),

ε

mN

p |lnε| if m

=NN(p––),

ε

mN

p if m

>NN(p––).

Proof Repeating the argument similar to (.), we compute

≤K(φ)≤B

(p–)N/p

ε

mN p rN+

m

p––dr+B

ε–

N(p–) (N–)p ε

mN

p rNNp––m– dr

+B

(p–)N/p

ε

mN p rN+

m

p––dr+B

N(p–) (N–)p

ε

mN

p rNNp––m–dr

=λ(ε) +λ(ε).

Secondly, we analyze howψ(u) aﬀectsK(ψ). The proof is similar to the above, and we omit it.

Lemma . Supposeψ(u)deﬁned in Theorem.satisﬁes

(B) There are positive numbersA,Bandqpsuch thatψ(u)≥B|u|qwhenuA.

Then we have K(ψ)≥λ(ε)asε→+,where

λ(ε) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(Np)q

(p–)pN if q<N(p–) Np ,

CεN(–pp)|lnε| if q=N(p–) Np ,

N(–p)

p if q>N(p–) Np ,

C is a positive constant.

(13)

(B) There exist positive numbersA,Bandq<p*such thatψ(u)B|u|qifuA.

Then K(ψ)≥λ(ε)asε→+,where

λ(ε) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

CεN(–pp) if q<N(p–) Np ,

N(–p)

p |lnε| if q=N(p–) Np ,

(pN)q

p if q>N(p–) Np ,

C is a positive constant.

In the following, we can utilize Theorem . and Lemmas .-. to prove the following results about some concrete problems. The proof is trivial and we omit it.

Corollary . Assume a(ξ) = ( +|ξ|)p– ξ,pin problem(.),and assumptions

(B)-(B), (C)hold,ψ(r)deﬁned in assumption(B)satisﬁes

(B) lim

ε→+

K(ψ)

λ(ε)

= +∞, whereλ(ε) =

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

εN+

(Np)

p(p–) if p>  –

N,

εN+

(Np)

p(p–)|lnε| if p=  –

N,

εN(p–)

(N–)p if p<  –N.

Then problem(.)possesses a nontrivial solution.

Proof Takem=  andm=p–  in Lemma ., and we can deduce the conclusion.

Example . Next, we consider the following equation:

⎧ ⎨ ⎩

–div(( +|∇u|)p– ∇u) =c|u|p*–u+k(x)|u|q–u+g(x,u), x,

u= , x. (.)

Corollary . Suppose the parameters≤p<q<p*and c> ,the functions k(x)C()

with k(x)k*> ,and g(x,u)C(×R)with g(x, ) = ,g(x,u)u.Moreover,

q>

⎧ ⎨ ⎩

p* N

N– if≤p≤ – 

N,

p*

p– if p≥ – 

N,

lim

u→∞supx∈

|g(x,u)| |u|p*– = ,

lim sup

u→ sup

x∈

g(x,u) |u|p–u<S.

Then problem(.)possesses a positive solution in C,α() (α(, )).

Example . Next, we consider the following equation:

⎧ ⎨ ⎩

–div(|∇u|p–u+|∇u|α+p–

+|∇u|p ∇u) =c|u|p

*–

u+k(x)|u|q–u+g(x,u), x,

(14)

Corollary . Suppose the parameters≤α<p and q<p*,c> ,the functions k(x) C()with k(x)k*> ,and g(x,u)C(×R)with g(x, ) = ,g(x,u)u.Moreover,

q>

⎧ ⎨ ⎩

p* N

N– ifα

N(p–) (N–),

Np ifαN(p–)

(N–),

lim

u→∞supx∈

|g(x,u)|

|u|p*– = , lim sup

u→ sup

x∈

g(x,u) |u|p–u<S.

Then problem(.)possesses a positive solution in C,α() (α(, )).

Proof Lettingm=  andm=αin Lemma ., and combining Lemma ., Lemma .

and Theorem ., we can derive the conclusion.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZY brought out the problem and gave the proof of the existence of a nontrivial solution. GD suggested that the generalized C-C principle could be applied to this problem and proved the non-trivial solution could be a positive solution. HY improved the regularity of the solution and checked all of the proof.

Author details

1School of Mathematics Science, South China Normal University, Guangzhou, 510631, China.2School of Mathematics and Computer Science, Guangdong University of Business Studies, Guangzhou, 510320, China.

Acknowledgements

The project is supported by National Natural Science Foundation of China (No. 11271143, 10901060, 10971073), the Natural Science Foundation of Zhejiang Province (No. Y6110775, Y6110789).

Received: 14 May 2012 Accepted: 13 September 2012 Published: 2 October 2012 References

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References

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